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Claim. For every infinite set $A$ with no distinct $a,b,c\in A$, $b,c>a$, $a\mid b+c$ (a good set in the formal-conjectures sense), $\sum_{n\in A}1/n<\infty$: the third question of Problem 12 answered yes. The claim's summary describes a block decomposition theorem: every good set splits into blocks with a multiplicative gap, $3a<b$ for $a$ in one block and $b$ in the next, which the goodness condition together with a Chinese-remainder structure is said to force, so the elements grow at least exponentially and the reciprocal sum converges by comparison with a geometric series; the summary presents this as the inverse theorem that the thread's barrier remark of 9 April 2026 anticipated. The claimants declare the proof checked by the Lean 4.27 kernel with one declared axiom, which they call oracle-verified and numerically checked over many configurations. The tab names the system as Libertas Superintelligence, described there as open-weight models with white-box reasoning, and the notes say it produced the discovery, the Lean verification and the write-up end to end. The eight-page write-up was sent to the site's curator and not posted; the public record of the claim on 2026-10-07 was the summary, the repository's listing and the two comments; apart from the declared axiom, shown false below, nothing in it has been checked.

Submission note. Posted to erdosproblems.com as a proof claim by Philip Ndikum, Serge Ndikum (account philipndikum) on 30 July 2026, giving "Libertas Superintelligence © (open-weight models, white-box reasoning)" as the AI used:

We resolve Part (iii) in the affirmative: for every good set A, the reciprocal sum Σ 1/n converges. The proof introduces a block decomposition theorem showing that any good set partitions into blocks B(i) with a cross-block multiplicative gap of 3a < b. This gap is forced by the definition of goodness together with the Chinese Remainder Theorem architecture. Consequently block elements grow at least exponentially, making the reciprocal sum converge by comparison with a geometric series. This is an inverse theorem in the sense predicted by Terence Tao (his April 9, 2026 comment on this thread). It proves that congruence constructions are nearly optimal — every good set must have this block structure. The proof is verified by the Lean 4.27 kernel with zero gaps and one oracle-verified axiom (numerically checked across 10,000+ configurations, following the established precedent of Problem #659). Github link: https://github.com/libertas-technology-group/Libertas-Erdos-Solutions Notes: Our system, Libertas Superintelligence ©, produces end-to-end solutions - discovery, Lean verification, and LaTeX paper production. It has been built for enterprise applications in regulated industries. We were inspired by Terence Tao's public talks on AI and mathematics, and decided to stress-test our system against open problems. The proof was an unexpected result of that testing. We are grateful to Dr. Bloom for maintaining erdosproblems.com — an invaluable resource and to Terence Tao for his thoughtful framework for AI-assisted mathematics. We do not have expertise in pure mathematics and warmly welcome expert feedback. The system produces white-box explanations of its reasoning — world model documents are available to any mathematician who wishes to review them. The PDF paper (8 pages) has been sent to Dr. Bloom directly.

Covers. The third question only. The first two are answered by the pending claim on the DeepMind claim page.

Standing. Rejected. The site's label is OPEN and its commentary, last edited 8 April 2026, does not mention the claim; the tab's notice says that appearance there is no guarantee of correctness. Two comments of 30 July 2026: one asks for a summary of the block decomposition lemma's proof; the other points at the declared axiom growth_ineq and reports that GPT 5.6 Sol, which the commenter consulted, did not find it correct. The axiom states that for every good set $A$ and every $i\ge0$ the $(S_i+1)$-th smallest element of $A$ is at least $P_i$, where $S_i=\sum_{j<i}\lfloor\sqrt{P_{j+1}}\rfloor$ and $P_i=10V_iY_i+C_i$, with $Y_i=3^{(i+20)^3}$, is the start of the $i$-th block of DeepMind's construction for the first question. It is false: at $i=0$ it says that every good set has least element at least $P_0>3^{8000}$, while the squares of the primes $p\equiv3\pmod4$ form a good set with least element $9$ (the 1970 example of Erdős and Sárközy, stated in formal-conjectures as isGood_example). The development's theorem summable_of_isGood rests on this axiom, so the Lean development proves nothing, and the claim is rejected; the third question stays open. No reply from the claimants, no refereed version and no outside review were found on 2026-10-07. The repository link is pinned to its head commit of 10 September 2026.

Depends on. No page of this wiki.